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Density-Functional Theory from h = 0 to 1

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Recent Classical and Quantum Applications to Aluminum Siting in Zeolites and the Freezing of Simple Fluids Shepard Smithline Cray Research, Inc., 655 E. Long Oak Drive, Eagan, M N 55121

Quantum and classical density functional theory have become important tools for describing many body phenomena in physics and chemistry. In this paper we review density functional theory, emphasizing the quantum and classical connections of the two theories. We show that both versions of the theory write the density functional as the sum of an ideal term and a term arising from inter-particle interactions. By differentiating the functional with respect to the density, it is straightforward to derive an equation that minimizes the functional. In the application of the quantum theory to electronic structure, the value of the functional at the minumum is the energy, while in the classical theory, the minimum value of the functional is the free energy or grand potential, depending on the thermodynamic conditions at hand. In addition, two examples of the theory are presented. The first, an electronic structure application, is to aluminum siting in zeolites, while the second, an application of the classical theory, is to the freezing of simple fluids. The electronic structure theory is able to predict the optimal location of aluminum in the zeolite cage, while the classical theory, despite some notable successes, is, on the whole, less successful in describing the liquid- solid transition. The reasons for the apparent shortcomings of the classical theory are discussed. Finally, it is speculated that the weighted density formalism, because it provides a means for constructing non-local functionals, might provide a framework for deriving improved functionals which is important for the further development of the theory.

Density functional theory has become ubiquitous in physics and chemistry. The theory has its origins in the Hohenberg-Kohn theorems which apply to quantum and classical systems (1). This allows the theory to be applied to a wide range of phenomena, ranging from nucleation in liquid-vapor systems (2) to the electronic structure of atoms and molecules (1). This chapter discusses density functional theory, emphasizing some of the parallels between the classical and quantum versions of the theory. In addition, 0097-6156/96/0629-0311$15.00/0 © 1996 American Chemical Society

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

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312

C H E M I C A L APPLICATIONS O F DENSITY-FUNCTIONAL T H E O R Y

we present two applications of the theory, one to aluminum siting in zeolite structures (3) and the other to the freezing of simple liquids (4). The paper is organized as follows. In section I I , we discuss some of the general quantum-classical connections in density functional theory, paying particular attention to the Hohenberg-Kohn theorems which form the foundation of density functional theory (1). Section I I I reviews the Kohn-Sham equations of electronic structure theory ι and applies them to a zeolite cluster, which models a heterogenous catalyst used in hydrocarbon separation (3). The fourth section shows how the classical theory can be applied to the freezing of simple fluids and is used to study the transition in a simple hard sphere fluid. Section V summarizes the weighted density functional formalism (5), describing how this idea can be used in quantum and classical systems to construct improved functionals. Finally, in section V I we conclude. Quantum-Classical Connections Two theorems proved by Hohenberg and Kohn provide the formal basis of density functional theorm (6),(7). The first theorem states that the external potential uniquely specifies the density of the system and vice versa: the density specifies the external potential of the system. Here density can refer to a quantum or classical density at constant particle number, temperature, and volume (canonical ensemble), or constant chemical potential, temperature and volume (grand canonical ensemble). The first Hohenberg-Kohn Theorem implies that once the density p is known everything that can be known is known about the system. For instance, in electronic structure theory, once the positions of the nucleii are given (the external potential in this case), the first Hohenberg-Kohn Theorem tells us that the electron density is determined. Since the density fixes the total number of electrons, we can write the complete Ν electron Hamiltonian and solve, in principle, for the many body wavefunction. Analogously, in the classical theory, fixing the external potential allows us to specify the canonical distribution function for any given system which has a particular interaction potential. The distribution function, in turn, allows us to calculate the equilibrium average of any observable. The second Hohenberg-Kohn theorem provides the theoretical underpinning for detemiining the density. It states for a given interaction potential and external potential V (r) ext

there exists an energy functional F[p] which is minimized when p = p ^ . Moreover, at the minimum, this functional equals the energy, free energy, or grand potential, depending on the thermodynamic conditions at hand. The proofs of the Hohenberg-Kohn theorems are well known (1), so we do not present them here. Both the classical and quantum proofs rely on the functional Fj/^Tr/U+p'V)

(D

which has the property that when f = f , the equilibrium distribution function, F[f] takes on its lowest value. This can be proved simply by considering the difference F[f] - F[f ] and showing that this difference is greater than zero for all f * f using a Gibbs inequality (8). Since the proof uses the standard properties of the trace, the trace may be a classical canonical trace, the classical grand canonical trace, or their quantum counterparts. Furthermore, the proofs utilize the fact that the equilibrium distribution function is the exponential of an energy divided by the trace of f, so f may refer to the classical or quantum distribution function. The zero temperature quantum case, corresponding to the conditions under which most quantum chemistry calculations are performed, is a special case of 1. As 0

0

0

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

22.

SMITHLINE

Density-Functional

313

Theory from 7i = 0 to 1

β - oo, F becomes the energy F-E=

. N i

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According to the variational theorem, the energy takes on its lowest value when ΨΝΪ = ΨΝΪ » ^ e ground state wavefunction. Thus, the variational theorem play the role of the Gibbs function at zero temperature and we can make the following analogies: 1. Energy 2. The expectation value 3. The ground state wavefunction ψ 4. The variational theorem

Ν ί

Free energy Trace The distribution function f The Gibbs theorem

Clearly, the zero temperature Hohenberg-Kohn theorems are closely related to their finite temperature cousins (1). Functionals are generally constructed by writing them as the sum of two parts, an ideal term, which ignores inter-particle interactions and is specified exactly, plus a term which accounts for the interaction and is known approximately. For instance, in electronic structure theory, the reference system is a collection of non-interacting electrons (1) and one writes E[P] = T [P]+ 8

"correction

(3)

where T [ p ] is the kinetic energy of a system of non-interacting electrons and the "correction" is J[p] + E [ p ] , J[p] being the classical coulomb potential and E [p] being the exchange correlation potential. This last term includes the quantum exchange terms left out of J[p] and the kinetic energy corrections to T [p]. We show below that 8

xc

xc

8

J[p] + E [p] plays the role of a mean field potential which generates the density. Similarly, for the classical case one often writes (8) xc

p[p]=*ι«[p]+φ[p]·

w

The expression F is the free energy of a non-interacting system. In applications to simple fluids, the reference system a monatomic gas. The second term in equation 4, φ [p ] , is the contribution to the free energy due to interactions and like J[p] + E [ p ] , gives rise to an effective one body potential which determines the density. Interestingly, not all density functional theories construct functionals by writing the functional as an ideal plus interacting piece. The geometric measures theory of Rosenfeld is one such example. In effect, it writes the free energy functional by interpolating between low and high density limits. We describe this theory in connection to hard sphere freezing in section IV. ideai

xc

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

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314

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

22. SMITHLINE

Density-Functional

315

Theory from h = 0 to 1

Electronic Structure Application

Kohn-Sham Theory. Almost all electronic structure calculations using density functional theory are carried out within the Kohn-Sham formalism (1). In this section we briefly review this approach and show how we can apply the formalism to an important problem in catalysis: the relative energetics of aluminum siting in zeolites (1). The Kohn-Sham method starts with the functional E[p] = T [p] + J[p] + Exc[p].

(5)

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e

By writing the density as p = £ [ y ^ a n d differentiating equation 5 with respect to the i

density, subject to the constraint that the Kohn-Sham orbitals one can derive the Kohn-Sham equations (-yVi +V 2

t f

)

V i

=e

remain orthonormal,

i V i

(6)

where t is a Lagrange multiplier resulting from the constraints on the orbitals and is the effective one body potential that generates the density. The exchange-correlation potential, which is included in the effective potential, is often computed using the local density approximation (LDA). LDA breaks space into small regions of nearly constant density. Since the exchange correlation energy for a uniform electron gas is known, the exchange correlation energy for the entire system can be computed by quadrature. The LDA approximation does surprisingly well, and is discussed below and by other authors in this volume. {

Application to Zeolites. We illustrate quantum density functional theory by studying zeolites (3). Zeolites are important catalysts, particularly in the petroleum industry, where they play a critical role in the catalytic cracking of large hydrocarbon molecules. The catalytic activity of zeolites is known to depend largely on their acidic properties which arise when aluminum atoms are replaced by silicon atoms. Consequently, it is of interest to know the location of aluminum atoms in the zeolite structure. Ab initio computations have predicted geometries and energies of a large number of compounds, but catalytic compounds have generally been too complex to study by such methods. To overcome these computational limitations, semi-empirical orbital calculations can be used to calculate the preferred aluminum position; however these methods require the selection of various parameters whose accuracy has not been determined for these compounds. As a result, we were lead to investigate the distribution of A l in mordenite using density functional theory. Calculations on two different clusters were performed: (i) A 39 atom cluster with formula S17O20H12 (ii) A 75 atom cluster, S114O39H22 in which the Si was replaced by A l at four different locations, known as Τ sites, and the relative energetics of the substitution was determined. Figure 1 shows the locations of these sites. T l and T2 reside on rings with 5 tetrahedral atoms (eg. A l or Si) while T3 and T4 are four atom rings. In addition to choosing the cluster geometry, a method must be chosen to truncate the cluster in such a way which preserves valency and charge distribution. This was accomplished by adding hydrogen atoms to the cluster to terminate the bonds.

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

316

CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

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Bonds lengths of .98 A for O-H and 1.43 for Si-Η were used. The geometries were taken from the de-hydrated structures of Schlenker, Pluth and Smith (9). All calculations were performed with DGauss, a density functional program developed by Cray Research, as part of the UniChem quantum chemistry package.(9) Calculations were performed with a valence double-zeta basis set with polarization (DZVP); these basis sets are comparable in size to 6-31G*, but have been optimized for DFT. BeckePerdew corrections to the local density approximation were added self-consistently. Table I below shows that T2 is predicted to be the most stable site (3). Table. I The relative energies of the different clusters for the four different Τ sites. All energies are measured in kcal/mole. Cluster 39 atom cluster 74 atom cluster

T l Site

T2Site

T3Site

T4site

24.6

0.0

11.3

13.9

10.10

0.0

4.12

16.18

Our prediction that T2 is the preferred site is consistent with experiments (10).(11), though some ambiguity remains. Bodart et. al. found that their experimental results are consistent with assuming that T3 and T4 are preferentially occupied, a result which was confirmed by previous computational studies on smaller clusters using Hartree Fock theory (12). In contrast, Itabishi drew different predictions from adsorption and NMR studies on synthetic mordenites. They argued that T2 site is a plausible location of Al which is consistent with our results. One effect that we have not completely investigated is the role that crystal relaxation plays in determining the relative ordering of the various sites. We are currently investigating this effect by allowing the aluminum and the nearest neighbor oxygen atoms to relax in the 74 atom cluster. Our preliminary results indicate that T2 remains the lowest energy site. We hope to report the results of this calculation in a future work. Classical Statistical Mechanics Application In contrast to quantum density functional theory, which is most often applied to questions of electronic structure, the classical theory is applied to a much wider variety of systems, ranging from liquid - vapor nucleation to polymeric systems. The intermolecular interactions in these systems are far more complex than in electronic structure problems, and because one usually does not know the exact Hamiltonian for these systems, relatively simple models are used to construct functionals. The results of these calculations are then compared to computer simulations. While the comparison to simulation may be quantitative, the real value of these theories is the qualitative understanding they often provide of actual many body systems. Here we apply the classical theory to the freezing of simple liquids. Freezing Theory. The starting point for developing a theory of first order phase transitions, such as freezing, is the Legendre transform of equation 11, or the grand potential functional, Ω[p]= where

u

Jdfp(r)V

e x t

(f)

+ Ρ^[p(τ)]-φ[p(?)]-μ

JdFpOD

(7)

is the chemical potential Here the reference system is an ideal monatomic fluid

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

22. SMITHLINE

Density-Functional Theoryfrom7i = 0 to 1

317

whose free energy is given by a simple analytic expression. Note, that working in the grand ensemble allows us to conveniently equate the chemical potential of the liquid and solid, one of the conditions which must be satisfied at the transition point. Differentiating equation 7 with respect to p (f ), and setting this functional derivative to zero yields p(F) = λ " 6 χ ρ [ β ( μ + V (r) + C[p(f)])

(8)

0[p(Τ)] = δφ/δρ.

(9)

3

ext

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where

Note, that equation 20 shows explicitly that the density is determined by an effective one body potential, analogous to ν in Kohn-Sham theory. It is straightforward to use this approach to study the freezing transition.(13) The derivation is given in reference 14. The final result is Λ

1η[ρ (?)/ρ] = / α Τ Ο [ τ ; μ 8

ί

5

,Τ ]-0[μ ,Τ,]-ε δ

(10)

ί

where C[f;μ ,TJ-C[μ ,TJ = JdTc^(ff-FI)[p (0-p ][p (f)-p ] s

L

s

L

s

L

+ J d 7 d 7 c \ ( l f - F I , l ? - r i ) [ p ( f ) - p J [ p ( f ) - p J [ p ( r " ) ~ p J + ... s

s

s

Equation 9 is the fundamental equation of the freezing theory and admits nontrivial solutions for a set of chemical potentials and temperatures. The solution corresponding to the freezing point is identified as that solution where the pressures of the two phases, calculated to the same order in perturbation theory, are equal. When the equality of pressures is satisfied, the solutions of 9 are guaranteed to generate a thermodynamically consistent transition point, since 9 was derived assuming that the temperatures and chemical potentials of the two phases are equal. Equation 9 is analogous to equation 6 in Kohn-Sham theory, as both arise by minimizing a functional with respect to a density subject to a constraint - the perfect crystal constraint in classical theory and the orthonormality of the Kohn-Sham orbitals in the quantum theory. Just as in Kohn-Sham theory, equation 9 is solved by expanding p(f) in a basis, such as Gaussians centered on lattice sites at the solid positions, R r

p(τ) = Σ ( α / π )

3/2

βχρ[-α(Γ-^) ] 2

(12)

i

or trigonometric functions,

ρω=Σ μ , / '

7

(13)

n=0

summed over reciprocal lattice vectors k . The particular lattice vectors or reciprocal n

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

318

lattice vectors are chosen to describe the symmetry of the particular solid one wishes to study (eg. fee, hep). The order parameters ( α or μ ) are determined by substituting equations 12 or 13 into 9 and solving the resulting algebraic equation. Equation 12 restricts the solid to isotropic and harmonic oscillations about the lattice sites while 13 lifts this approximation, being a general representation for a periodic structure. Expanding the solid density in a basis illustrates one of the central assumptions of the theory. One must assume the symmetry of the solid; the theory does not predict the symmetry of the solid, though it does predict the magnitude of the lattice vectors. Currently, there is no a priori way of addressing the issue of spontaneous symmetry breaking in classical density functional theory. Another major assumption concerns expansion 11. Even if the expansion (4) is carried out to infinite order, a functional Taylor expansion like 11 suppresses fluctuations by prohibiting configurations in which some regions are liquid-like and others to be solid-like. This assumption is known as the homogeneity approximation and is central in all mean field theories of phase transitions. Like the assumption of spontaneous symmetry breaking, we know of no way of addressing this approximation.

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η

Application to the Freezing of H a r d Spheres. The freezing theory has been applied to a wide variety of problems, ranging from systems with purely repulsive forces, such as hard spheres and inverse power potentials, to systems with attractive forces, such as Lennard-Jones fluids and the one-component plasma, to more complex systems such as water and polymers (4). The results are some what inconsistent. For hard spheres the theory performs well, yielding a reasonably accurate freezing density and the density change on freezing, for instance (4). For moderately complex systems, such as inverse power potentials, the theory does less well, failing for instance to predict the bec phase ε ( σ / Γ ) , η * 6 where ε and σ measure the strength and characteristic length of the interaction (14). For more complex fluids, the theory exhibits "re-entrant" behavior, successfully predicting, for instance, the freezing of water (15) and polyethylene (16). Given this somewhat inconsistent performance, we re-examined the theory. One of the critical assumptions of the classical theory is the truncation of expansion 11 at second order. While the higher order terms are not thought to be small, it is often assumed the integral of them times the corresponding density difference is. This assumption is usually not checked because the evaluation of these terms is very difficult Recently, however, a theory for constructing free energy functionals has been proposed and allows the higher order terms to be computed by functional differentiation of F[p(r)] (17). When the triplet term is computed for hard spheres, it is found to agree quite well with computer simulations, and thus, it is natural to see how this term affects die results of the freezing theory. Before we present these results, we begin with a brief derivation of this Rosenfeld's free energy functional. Our discussion closely follows reference 18, and the reader is encouraged to consult it for more details. Rosenfeld postulated the following general form for the excess free energy functional (that part over and above the ideal contribution) as, η

F J t p . f f ) } ] = J d f «{n Cf) n 0f)}] = J d f φ[{η (Τ)}], 3

3

e

f

q

α

(14)

where n (f) is a weighted density a

n (f)= J d r a

3

P i

(F')û)(r-r),

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

(15)

22.

SMITHLINE

Density-Functional

319

Theory from S = Oto 1

where Pj(f) is the density of species i.

Rosenfeld gives the explicit forms for

cof(F-F). They depend on the geometrical properties of the spheres and consist of vector and scalar functions. Equation 14 allows us to write the excess pressure function, Π [ { η } ] , and chemical potentials, μ and μ , as α

βχ

Μ

Π=-φ+Ση θφ/θη α

(16)

α

α

μ ? = /ά ϊ'Σοφ/5η [{η (Τ)}]ωΓ(Τ-Γ ) 3

β

(Π)

γ

α

μ^ΐηίρ^λ ]

(18)

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3

Although it is probably not readily apparent at this point, the assumption that the excess free energy density is a functional of only the fundamental-measures weighted density, restricts the final form of φ [ { η } ] considerably. If we impose the exact relation for the α

μί->Ρν

uniform chemical potential , on our fundamental measure 4 for R ^ description of the non-uniform fluid, the following differential equation must be satisfied, 0 0

Π+ η =3φ/θη 0

(19)

3

This relation can be derived by observing that the scalar and vector weights respectively satisfy Jd'TVff^l.R,,

(20)

Jd FÔL>?(f)=0,

(21)

3

where Rj, Si, and Vi are the radius, surface area, and volume of sphere i. Consequently for homogeneous fluids, equation 17 can be re-written as μ"=δφ/δη +δφ/δη^ +δφ/δη 8 +δφ/δη ν 0

ί

2

ί

3

(22)

ί

Now since P / k T = Σ Pi» where P is the ideal pressure, then i d

id

f

P w

1= V,i Jd'îEp.ÔOfl-R.y^nR?)

. kT .

ψ

=V,n

0

(23)

sinceœ" isdefinedasœj (F-r)=6(lf l-r)/(4jcr ). Therefore, ,

>

2

PV,=V,{n +n} e

and imposing μ) - PVj yields,

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

(24)

320

C H E M I C A L APPLICATIONS O F DENSITY-FUNCTIONAL T H E O R Y

V n+V n = ln[p(ïU ]+

·

3

i

i

0

(25)

Upon dividing by Vj and letting Rj->«>, only δφ/δη survives, giving the scaled particle differential equation 19. Using 19 we can write equation 16 as, 3

-φ+Ση 3φ/3η + η = 3φ/θη α

α

0

(26)

3

α

Furthermore, the only positive integer power combinations which have units of V " (as required by the virial theorem) are Downloaded by STANFORD UNIV GREEN LIBR on October 9, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch022

1

n , n , n , ( n ) , fij.ii , n ( n V n ) ,

(27)

2

0

2

2

2

2

2

and therefore form a basis for φ, φ=ί η + ί 0

ο

n n +f

12

1

2

2 2 2

( n ) +f ïi .n + f 2

2

12

1

2

222

n (nVii ) . 2

(28)

2

The functions f are dimensionless, and for generality are allowed to depend on n , a dimensionless quantity. Substituting equation 28 into 26 results in five differential equations which allow the f s to be determined. The boundary conditions of the differential equations are chosen so that equation 28 reproduces the low density expression for the free energy density and the three particle diagram for c (r). As a result it can be shown that, 3

2

Φ=Φ. +

(29)

Φν

where φ = - η 1 η ( 1 - η ) + η η / ( 1 - η ) + 1/(24π)η /(1-η ) 3

8

0

3

1

2

3

2

(30)

2

3

and φ^ϋ^ϋ,/ίΙ-η,ί+Ι/ίδπ^ίη,.η,ΧίΙ-η,)

(31)

2

Now the m-th order direct correlation functions are given by c™, (T ï lf

ae

...f ) = m

δΡ [{p:(Τ)}]/κΤ " J ' ' βχ

(32) = Jd x 3

Σ {aV^n ...an Jœ (f -x)....œ° (? -x) a,

ai

a

m

1

m

If we fourier transform c™ ...j and note that in the uniform liquid d tyI d η ... d n m

Γ

ni

α

1

a

m

are

independent of position, then in k-space the direct correlation function is simply a linear combination of weight functions

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

22. SMITHLINE

Density-Functional

321

Theory from 7i = 0 to 1

φ ΰ = - Σ 3 φ / β ι ι 3 n ftr(k)©f(k) *β 2

(33)

p

β

Equation 33 can be inverted to yield the real space representation which can be shown to be equivalent to the Perçus-Yevick direct correlation function. Similarly for the triplet function we can write, c ( k , k , k ) = - Σ e ^ / a n a n a n ^ ( k ) û f ( k ) û ; ( k , ) 6 ( k + k + k 3 ) (34) 3

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1

2

e

3

p

1

2

l

2

«.β.γ

There are a total of 33 terms in expression 34, explicit expressions for which are given in Rosenfeld's paper. Our derivation of 34, which is in excellent agreement with computer simulation results and relies on the scaled particle differential equation, shows how one can construct a free energy functional that effectively interpolates between the low and high density results. Now, returning to the third order contribution to the free energy, we note that the triplet function's contribution to the free energy can be evaluated (18) as 4=

Σ M

c (k w

l f

k ,k ^^ 2

3

A

1

(35)

k

2

where the sum is over all reciprocal lattice vectors ί of the solid subject to the triangle {

condition: E = - ^ - £ . The coefficients μι^ are the fourier coefficients of a Gaussian 3

2

solid density. The results, shown in Table Π, indicate that the third order term is nearly as large as the second order term, and as a result, there is no a priori way of justifying truncation of equation 11 at second order. Table I I . The second and third order contribution to the free energy functional divided by kT^oOf fee hard spheres (of diameter σ) of average solid density p,& and gaussian width ασ as a function of liquid density, solid density and gaussian width: 3

a. p a = . 7 9 , ρ σ ' = .975, α σ ' = 180;

b. p a = .975, ρ σ ' = .975, α σ = 180;

C.

d. p a = .975, ρ σ ' = .975, α σ = 50

L

β

p G = .79, ρ σ ' = .79, α σ = 5 0 ; L

3

3

t

L

3

β

3

5

3

For comparison the results of Curtin-Ashcroft (reference 22) are also shown. Second-Order Terms

Third-Order Terms (us)

a

-3.8857

-1.6502

b

-3.2576

-4.2856

c

-1.5802

-1.5918

d

-2.1094

-1.588

Third-Order Terms (Curtin-Ashcroft) -1.841

These results strongly suggest that the apparent agreement between the second order theory freezing theory and computer simulation data is fortuitous in sense that one does not understand why truncating at second order should result in a good description of the liquid-solid transition. While we performed calculations only for hard spheres, we expect our results to be qualitatively correct for more realistic fluids which contain attractive as

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

322

CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

well as repulsive forces. The reason for the apparent failure of expansion equation 11 is explored in section V. Non-local Functionals One of the common assumptions of LDA theory and the freezing theory discussed above is that the free energy functional is a local functional of the density. That is, the density is evaluated at a point in space and from the value of the density at that point in space, the free energy is evaluated. However, we know quantum exchange is inherently non-local. For example within Hartree-Fock theory exchange operator K (ΐ ) is given by Downloaded by STANFORD UNIV GREEN LIBR on October 9, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch022

b

ι

(36) which is clearly non-local. Similarly, we expect the free energy functional for the classical freezing theory to have non-local character. The origin of the classical nonlocality can be understood from F=Jd rp(r)y(p(f)) 3

(37)

where Ψ(ρΟΟ) is the free energy per-particle in a homogeneous system. This expression for the free energy implies that a particle at r is only affected by particles around it in a given range of interaction. If the range of the particle interactions is smaller than the length scales over which the density varies, then it is often reasonable to break the system up into small pieces, each one of nearly constant density, evaluate the free energy of each piece as if it were part of a homogeneous system, and add them up (integrate them) to get the total free energy. This prescription is followed in LDA theory. However, this approach is of limited use in the liquid -solid transition. Since the density of the solid reaches large peak values, the value of the free energy functional of the corresponding homogeneous system, is almost certain to be quite different the free energy functional of the real system. In fact, the density may even be impossible to achieve, as is the case for hard spheres, when the density exceeds the close-packing limit. It is possible to imagine other schemes which employ a homogenous reference system to evaluate the free energy of an inhomogeneous system, but, if the density is not smooth on the relevant length scales, then a local density approximation is likely to break down. A fluid at a first order phase transition or up against a hard wall are clearly such cases. Thus, even if we could sum the series of equations 11 to beyond third order, it is not clear, a priori, that this would result in a good description of the liquid-solid transition, since expansion equation 11 is an inherently local approximation. Still the notion of using a homogeneous system to describe inhomogeneous systems is appealing, provided one can introduce non-locality into the functional. One way to construct a non-local functional is to use a weighting functional. This idea was first used in electronic structure density functional theory ,(19) and subsequently used by researchers in classical liquid theory (20). The essential idea is to replace p(f) in equation 7 by a weighted density, (38)

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

22.

SMITHLINE

Density-Functional

Theory from Λ = 0 to 1

323

Note that Rosenfeld's scalar weights are a special case of 38 for which we set

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w (r,r\p(r))=co(r,F)

(39)

There are two general ways of choosing the weight functions. One is to select them by a priori means, as done by Rosenfeld. The other is to "tailor" the weight function to reproduce some known properties of the homogeneous fluid. There are disadvantages to both techniques. If we adopt Rosenfeld's theory, then the weighted density cannot easily be interpreted as an effective local density due to the presence of the vector weight functions, as the weighted density now becomes a vector quantity not a simple scalarfunction. On the other hand, if we tailor the weight functions to mimic some property_of the homogeneous fluid then it is possible to derive a relation for w \r,F , p (r) j . For instance, if we use the full weight function then one can derive a differential equation which relates the weight function to the pair correlation function (21). Alternatively, if one is restricted to scalar weighted densities which are simple linear averages of the density (as in equation 15), one can derive weight functions, though they are non-trivial, as they involve derivatives of delta functions (22). The "tailored" functionals also have been applied to study the freezing of simple fluids (4) and while in some cases they give better answers, they still are not free from artifacts (4),(17). Interestingly enough, had we used the Rosenfeld weighting functions, and determined φ by tailoring it to reproduce the Perçus - Yevick direct correlation function instead of using the scaled particle differential equation, we would not have found Rosenfeld's vector component of φ and for certain values of the k vectors, the vector part of the triplet function contributes about half the value of the value of the total function. As a result of the somewhat arbitrary nature of deriving weighted density functionals, it is clear that the idea needs to be further refined before it can be used reliably to describe the liquid-solid transition. It should also be pointed out that the weighted density approximation was abandoned in electronic structure theory because other approaches proved to be more reliable (23). The original implementation of the weighted density idea used the randomphase approximation to compute the exchange-correlation functional for a homogeneous electron gas (21). However, if one could tabulate the exact pair function for a homogeneous electron gas, similar to the way one tabulates the exchange-correlation energy of the electron gas, then the RPA approximation could be lifted. As a result, the weighted density approximation, when combined with the various scaling relations used to construct the gradient corrected non-local functionals, might lead to new and more accurate models for the exchange-correlation energy. βχ

βχ

Conclusion

Density functional theory has its origins in the Hohenberg-Kohn theorems. Since these theorems apply equally well to quantum and classical systems, density functional theory can be applied to problems in quantum and classical mechanics. Given the theories' common origin, it is not surprising that there are many similarities between the quantum and classical versions of the theories, and in this paper we discussed some of these similarities. We showed that both the quantum and classical versions of the theory often write the appropriate energy functionals the sum of an ideal and an interaction term. By minimizing the functional with respect to the density, it is straightforward to derive an equation for the density where the density is determined by a mean field potential arising from the inter-particle interactions. Besides discussing quantum-classical analogies in density functional theory, we

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

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324

CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

illustrated the theory by presenting two applications, one to aluminum siting in zeolites and the other to the freezing of classical fluids. The quantum version of the theory is able to predict the optimal location of aluminum in the zeolite cage while the classical theory, despite some notable successes, is less successful in describing the liquid- solid transition. The apparent shortcomings of the classical theory reflect the inherent difficulty of developing a first principles theory for first order phase transitions. Indeed, we argued that a density functional theory of freezing probably requires a sophisticated non-local free energy functional. Unfortunately, it is not at all clear how to construct such a functional. One possibility is to use a weighted density formalism. Besides being potentially useful in classical theory, this approach might also be helpful in deriving newnon-local functionals for quantum density functional theory, although additional work needs to be done to further develop this idea. Nevertheless, whatever the outcome of future research into weighted densities, we are optimistic that, given the past successes of density functional theory, the theory will continue to be an exciting and useful formalism for describing many body phenomena, be they quantum or classical systems. Acknowledgements The author is happy to acknowledge the many chemists at Cray Research, including George Fitzgerald, Rich Graham, Chengthe Lee, and Eric Stahlberg, who taught him quantum density functional theory. The author also thanks Cray Research, Inc. for providing the computer resources used to carry out some of the calculations reported here.

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